Method for analytically obtaining closed form expressions for subsurface temperature depth distribution along with its error bounds

ABSTRACT

Analytical solutions to error bounds on the temperature depth distribution have been given in this invention. Solving the one dimensional steady state heat conduction equation for different sets of boundary conditions and radiogenic heat generation and incorporating Gaussian randomness in the thermal conductivity analytical closed form solutions to the mean and variance in the temperature depth distribution have been obtained. These closed form analytical solutions of mean and variance for the temperature field for different conditions have been used to compute and display the plot and results of the temperature depth profiles along with its error bounds. Quantifying the error statistics in the system output due to errors in the system input is very essential for a better evaluation of the system behavior. Earth Scientists involved in understanding the subsurface thermal structure relevant to geodynamical studies will benefit using these findings.

FIELD OF THE INVENTION

[0001] The present invention relates to analytical solution to errorbounds on the subsurface temperature depth distribution. This inventionhas a wide range of application in quantifying the subsurface thermalstate of the crust and has a wide range of applications. The method ofthe invention has helped in obtaining closed form expressions for thesubsurface temperature depth distribution along with its error bounds.The exact formulae are useful in better evaluating the thermal state andhave a wide range of applications in oil and natural gas prospecting,tectonic studies and mineral prospecting.

BACKGROUND OF THE INVENTION

[0002] This invention relates for obtaining and computing the subsurfacetemperature depth distribution along with its error bounds. The solutionhas been determined for the stochastic heat conduction equation byconsidering different sets of boundary conditions and radiogenic heatsources and incorporating randomness in the thermal conductivity. Inunderstanding the Earth thermal structure there are several questionswhich need clear answers. Many of the controlling parameters that definethe Earth's processes are not known with certainty. In such situationsthese controlling parameters can be defined in a stochastic frameworkand an average picture of the system behavior together with its errorbounds can be quantified.

[0003] The thermal structure of the Earth's crust is influenced by itsgeothermal parameters such as thermal conductivity, radiogenic heatsources and initial and boundary conditions. Basically two approaches ofmodeling are commonly used for the estimation of the subsurfacetemperature field. These are: (1) deterministic approach and (2) thestochastic approach. In the deterministic approach the subsurfacetemperature field is obtained assuming that the controlling thermalparameters are known with certainty. However, due to inhomogeneousnature of the Earth's interior some amount of uncertainty in theestimation of the geothermal parameters are bound to exist.Uncertainties in these parameters may arise from the inaccuracy ofmeasurements or lack of information about the parameters themselves.Such uncertainties in parameters are incorporated in the stochasticapproach and an average picture of the thermal field along with itsassociated error bounds are determined. To assess the properties of thesystem at a glance we need to obtain the mean value that gives theaverage picture and the variance or the standard deviation that is thevariability indicator which gives the errors associated with the systembehavior due to errors in the system input.

[0004] Subsurface temperatures are also seen to be very sensitive toperturbations in the input thermal parameters and hence several studieshave been carried out in quantifying the perturbations in thetemperatures and heat flow using stochastic analytical and randomsimulation techniques. Quantification of uncertainty in the heat flowusing a least squares inversion technique incorporating uncertainties inthe temperature and thermal conductivities has been done,Tectonophysics, Vol 121, 1985 by Vausser et al. The effect of variationin heat source on the surface heat flow has also been studied, JournalGeophysical Research, V 91, 1986, by Vasseur and Singh, GeophysicalResearch Letters, V14, 1987, by Nielsen. In most of the studies thestochastic heat equation has been solved using the small perturbationmethod. Using the small perturbation method the heat conduction equationhas been solved by incorporating uncertainties in the heat sources andthe mean temperature field along with its error bounds have beenobtained, Geophysical Journal International, 135, 1998, by Srivastavaand Singh. The random simulation method has also been used to model thethermal structure incorporating uncertainties in the controlling thermalparameters, Tectonophysics, V156, 1988 by Royer and Danis, Marine andPetroleum Geology, V 14, 1997, by Gallagher et al, Tectonophysics, V306, 1999a,b, by Jokinen and Kukkonen. This numerical modeling is veryuseful in studying the nonlinear problems but sometimes simple 1-Danalytical solution to the mean behavior and its associated error boundsis very useful in quantifying the uncertainty. The stochasticdifferential equations in other fields are now being solved by yetanother approach called the decomposition method, Journal of Hydrology,V 169, 1995, by Serrano. In a recent study using this new approach thestochastic heat equation has been solved incorporating uncertainties inthe thermal conductivity where the solution to the temperature field isobtained using a series expansion method, Geophysical JournalInternational, V 138, 1999, by Srivastava and Singh. The thermalconductivity is considered to be a random parameter with a knownGaussian colored noise correlation structure.

[0005] In this invention the stochastic solution to the mean andvariance in the temperature field for a different set of boundaryconditions and different radiogenic heat source function has beenobtained following the procedure of Geophysical Journal International, V138, 1999, by Srivastava and Singh. The expressions for mean andvariance in temperature depth distribution for different heat sourcesand boundary conditions have been obtained and used to compute and plotthe subsurface thermal field along with its error bounds.

OBJECTS OF THE INVENTION

[0006] The main object of the invention is to provide analyticalsolution to error bounds on the subsurface temperature depthdistribution which obviates the drawbacks detailed above

[0007] Another object of the invention is to provide an efficient methodfor obtaining closed for solution to error bounds on temperature depthdistribution for different set of boundary conditions.

[0008] Yet another object of the invention is to provide forquantification of subsurface temperature depth distribution and itserror bounds for known Gaussian thermal conductivity structure.

BRIEF DESCRIPTIONS OF THE DRAWINGS

[0009] The attached figures (FIG. 2-7) are the plots of mean temperature±1 standard deviation for different values of the controlling thermalparameters.

[0010]FIG. 1 is a flow sheet depicting the method of the invention.

[0011]FIG. 2 Plot of mean temperature ±1 standard deviation forcondition 1 when no heat source is considered and the boundaryconditions employed are constant surface temperature and constantsurface heat flow.

[0012]FIG. 3 Plot of mean temperature ±1 standard deviation forcondition 2 when no heat source is considered and the boundaryconditions employed are constant surface temperature and constant basalheat flow.

[0013]FIG. 4 Plot of mean temperature ±1 standard deviation forcondition 3 when a constant heat source is considered and the boundaryconditions employed are constant surface temperature and constantsurface heat flow.

[0014]FIG. 5 Plot of mean temperature ±1 standard deviation forcondition 4 when a constant heat source is considered and the boundaryconditions employed are constant surface temperature and constant basalheat flow.

[0015]FIG. 6 Plot of mean temperature ±1 standard deviation forcondition 5 when an exponential heat source is considered and theboundary conditions employed are constant surface temperature andconstant surface heat flow.

[0016]FIG. 7 Plot of mean temperature ±1 standard deviation forcondition 6 when an exponential heat source is considered and theboundary conditions employed are constant surface temperature andconstant basal heat flow.

SUMMARY OF THE INVENTION

[0017] Accordingly the invention provides an analytical solution toerror bounds on the subsurface temperature depth distribution, whichcomprises a method of solving the heat conduction equation incorporatingGaussian uncertainties in the thermal conductivity.

[0018] In an embodiment of the present invention the stochastic heatconduction equation has been solved using a series expansion method toobtain the closed form solution to the mean and variance in thetemperature depth distribution. Simple deterministic solution to theproblem is not sufficient and quantifying the errors in the systemoutput due to errors in the input parameters is very essential. Theseerrors bounds are very important for a better evaluation of thesubsurface thermal structure.

[0019] In another embodiment of the present invention the expression formean temperature and the variance in temperature are obtained for sixdifferent set of prescribed boundary conditions.

DETAILED DESCRIPTION OF THE INVENTION

[0020] The present invention deals with the solution to the governingstochastic heat conduction equation to obtain the mean and variance inthe temperature fields as shown in the flow chart. The heat conductionequation with random thermal conductivity is expressed as$\begin{matrix}{{\frac{\quad}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {- {A(z)}}} & (1)\end{matrix}$

[0021] where

[0022] T is the temperature (° C.)

[0023] A(z) is the radiogenic heat source (μW/m³),

[0024] K(z)={overscore (K)}+K′(z) is the thermal conductivity (W/m ° C.)which is expressed as a sum of a deterministic component and a randomcomponent

[0025] K′(z) is the random component with mean zero and a Gaussiancolored noise correlation structure represented by

E(K′(z))=0  (2)

E(K′(z ₁)K′(z ₂))=σ_(K) ² e ^(−p|z) ^(₁) ^(−z) ^(₂) ^(|)  (3)

[0026] where

[0027] σ_(K) ² is the variance is thermal conductivity (W/m ° C.)²

[0028] ρ is the correlation decay parameter m⁻¹ (or 1/ρ is thecorrelation length scale)

[0029] z₁ and Z₂ are the depths (m)

[0030] Following the procedure of given in Geophysical J International,V 138, 1999 by Srivastava and Singh, the solution to mean temperatureand its standard deviation has been obtained for three conditions ofheat sources (1) Zero (A(z)=0) (2) Constant (A(z)=A) and (3)Exponentially decreasing with depth (A(z)=A₀e^(−z/D)) and

[0031] Associated boundary conditions are defined by

[0032] Type(i) Boundary Condition:

[0033] Constant Surface Temperature

T=T₀ at z=0  (4)

[0034] Surface heat flow Q_(S) (mW/m²) $\begin{matrix}{{\overset{\_}{K}\frac{T}{z}} = {{Q_{s}\quad {at}\quad z} = 0}} & (5)\end{matrix}$

[0035] Type (ii) Boundary Condition:

[0036] Constant Surface Temperature

T=T₀ at z=0  (6)

[0037] Basal heat flow Q_(B) (mW/m²) $\begin{matrix}{{\overset{\_}{K}\frac{T}{z}} = {{Q_{B}\quad {at}\quad z} = L}} & (7)\end{matrix}$

[0038] The standard deviation, which is a measure of error in the systemoutput, is obtained by taking the square root of the variance. Thesolutions to different conditions obtained have been given below.

[0039] Condition 1: When no heat source is considered and the B.C usedare the surface temperature and surface heat flow

[0040] The governing heat conduction equation without heat source termis $\begin{matrix}{{\frac{\quad}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = 0} & (8)\end{matrix}$

[0041] with constant surface temperature and constant surface heat flowas boundary conditions

[0042] (Type (i))

[0043] Solution

[0044] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{Q_{s}}{\overset{\_}{K}}z}}}} & (9)\end{matrix}$

[0045] Variance in temperature

σ_(T) ² =c1×Term1  (10)

[0046] where

c1=2C _(K) ² Q _(S) ² /{overscore (K)} ²

[0047] and${Term1} = {{\rho \frac{z^{3}}{3}} - \frac{z^{2}}{2} + {\frac{1}{\rho^{2}}\left( {1 - ^{{- \rho}\quad z}} \right)} - {\frac{z}{\rho}^{{- \rho}\quad z}}}$

[0048] where C_(K)=σ_(K)/{overscore (K)} is the coefficient ofvariability in the thermal conductivity.

[0049] Condition 2: When no heat source is considered and the B.C usedare surface temperature and heat flow at the base of the model(Q_(S)=Q_(B))

[0050] The governing heat conduction equation without heat source termis $\begin{matrix}{{\frac{\quad}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = 0} & (11)\end{matrix}$

[0051] with constant surface temperature and constant basal heat flow asboundary conditions

[0052] (Type (ii))

[0053] Solution

[0054] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{Q_{B}}{\overset{\_}{K}}z}}}} & (12)\end{matrix}$

[0055] Variance in Temperature

σ_(T) ² =c1×Term1  (13)

[0056] where

c1=2C _(K) ² Q _(B) ² /{overscore (K)} ²

[0057] and

[0058] Term 1 is same as given in conditions.

[0059] Condition 3: When constant heat source is considered and the B.Cused are the surface temperature and surface heat flow

[0060] The governing heat conduction equation with constant heat sourceterm is $\begin{matrix}{{\frac{\quad}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {- A}} & (14)\end{matrix}$

[0061] with constant surface temperature and constant surface heat flowas boundary conditions

[0062] (Type (i))

[0063] Solution

[0064] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{Q_{s}}{\overset{\_}{K}}z} - \frac{A\quad z^{2}}{2\quad K}}}} & (15)\end{matrix}$

[0065] Variance in Temperature

σ_(T) ² =c1×Term1+c2×Term2+c3×Term3+c4×Term4  (16)

[0066] where

[0067] c1=C_(K) ²(A−ρQ_(S))²/{overscore (K)}²

[0068] c2=C_(K) ²Aρ(A−ρQ_(S))/{overscore (K)}²

[0069] c3=c2

[0070] c4=C_(K) ²ρ²A²/{overscore (K)}²

[0071] and $\begin{matrix}{{Term1} = {\frac{1}{\rho^{2}}\left( {{\rho \quad \frac{2z^{3}}{3}} + {\frac{^{{- \rho}\quad z}}{\rho^{2}}\left( {{{- \rho}\quad z} - 1 + ^{\rho \quad z}} \right)} - {\frac{\left( {{\rho \quad z} + 1} \right)}{\rho^{2}}\left( {{\rho \quad z} + ^{{- \rho}\quad z} - 1} \right)}} \right)}} \\{{Term2} = {\frac{\left( {{\rho \quad z} + 2} \right)}{\rho^{3}}\left( {\left( {\left( {{\rho \quad z} + ^{{- \rho}\quad z} - 1} \right)/\rho^{2}} \right) + {\rho \quad \frac{z^{3}}{2}} +} \right.}} \\{{\left. \quad {\frac{z^{2}}{2} - \frac{\rho \quad z^{3}}{3} - z^{2}} \right)\frac{z^{4}}{12\quad \rho}} + {\frac{\left( {{\rho^{2}z} - {2\quad \rho}} \right)}{\rho^{3}}\frac{z^{3}}{6}\quad \frac{z^{4}}{12\quad \rho}} +} \\\left. \quad {\frac{\left( {{\rho \quad z} - 2} \right)z^{2}}{2\quad \rho^{3}} - {\frac{\left( {{\rho \quad z} + 2} \right)}{\rho^{5}}\left( {{{- \left( {{\rho \quad z} + 1} \right)}^{{- \rho}\quad z}} + 1} \right)}} \right) \\{\left. {{Term3} = {{\frac{1}{\rho^{2}}\left( {\frac{\rho \quad z^{4}}{6} + {\frac{\left( {{\rho \quad z} + 1} \right)}{\rho^{2}}\left( {{\left( {{\rho \quad z} + 1} \right)^{{- \rho}\quad z}} - 1} \right)\frac{\left( {{\rho \quad z} - 2} \right)}{\rho}}} \right)} - {\rho \quad z^{2}^{{- \rho}\quad z}}}} \right) +} \\{\quad {\frac{\left( {{\rho \quad z} - 1 + ^{{- \rho}\quad z}} \right)}{\rho^{2}}\left( {\left( {z + \frac{2}{\rho}} \right) - \frac{z^{2}}{\rho}} \right)}} \\{{Term4} = {\frac{\left( {z + {2/\rho}} \right)}{\rho^{2}}\left( {\frac{\rho \quad z^{4}}{12} - \frac{z^{3}}{6} + {\left( {z^{2}^{{- \rho}\quad z}} \right)/\rho} +} \right.}} \\{\left. \quad {\left( {\left( {{{- \left( {{\rho \quad z} + 1} \right)}^{{- \rho}\quad z}} + 1} \right)\frac{\left( {z - {2/\rho}} \right)}{\rho^{2}}} \right) - \frac{z^{5}}{20\quad \rho}} \right) +} \\{\quad {{\frac{\left( {z + {2/\rho}} \right)}{\rho^{2}}\left( {{- \left( {{\rho \quad z} - 1} \right)} + ^{{- \rho}\quad z}} \right)\frac{\left( {{z\quad \rho} + 1} \right)z}{\rho^{2}}} +}} \\{\quad {{\left( {{\rho \quad z} + 1} \right){^{{- \rho}\quad z}\left( {\frac{z^{2}^{\rho \quad z}}{\rho} - {\frac{2}{\rho^{3}}\left( {{\left( {{\rho \quad z} - 1} \right)^{\rho \quad z}} + 1} \right)}} \right)}} +}} \\{\quad {\frac{\rho \quad z^{4}}{12} + \frac{z^{3}}{6} + {\frac{\left( {z + {2/\rho}} \right)}{\rho^{3}}\left( {{\left( {{\rho \quad z} - 1} \right)^{\rho \quad z}} + 1} \right)z^{2}^{{- \quad \rho}\quad z}} - \frac{z^{5}}{20\quad \rho} - \frac{z4}{\rho^{2}}}}\end{matrix}$

[0072] Condition 4: When constant heat source is considered and the B.Cused are the surface temperature and heat flow at the base of the model

[0073] The governing heat conduction equation with constant heat sourceterm is $\begin{matrix}{{\frac{}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {- A}} & (17)\end{matrix}$

[0074] with constant surface temperature and constant basal heat flow asboundary conditions

[0075] (Type (ii))

[0076] Solution

[0077] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{\left( {Q_{B} + {A*L}} \right)}{\overset{\_}{K}}z} - \frac{A\quad z^{2}}{2K}}}} & (18)\end{matrix}$

[0078] Variance in Temperature

σ_(T) ² =c1×Term1+c2×Term2+c3×Term3+c4×Term4  (19)

[0079] where

[0080] c1=C_(K) ²(A−ρ(Q_(S)+A*L))²/{overscore (K)}²

[0081] c2=C_(K) ²Aρ(A−ρ(Q_(B)+A*L))/{overscore (K)}²

[0082] c3=c2

[0083] c4=C_(K) ²ρ²A²/{overscore (K)}²

[0084] Term1, Term2, Term3 and Term4 are same as given in condition3.

[0085] Condition 5: When an exponential heat source function isconsidered and the B.C used are the surface temperature and surface heatflow

[0086] The governing heat conduction equation with an exponential heatsource term $\begin{matrix}{{\frac{}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {{- A_{0}}^{{- z}/D}}} & (20)\end{matrix}$

[0087] with constant surface temperature and constant surface heat flowas boundary conditions

[0088] (Type (i))

[0089] Solution

[0090] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{Q_{s}}{\overset{\_}{K}}z} + {\frac{A_{0}D^{2}}{\overset{\_}{K}}\left( {1 - \frac{z}{D} - ^{{- z}/D}} \right)}}}} & (21)\end{matrix}$

[0091] Variance in the Temperature

ρ_(T) ² =c1×Term1+c2×Term2+c3×Term3+c4×Term4  (22)

[0092] where the constants are

[0093] c1=C_(K) ²A₀ ²(1−ρD) ²/{overscore (K)}²

[0094] c2=C_(K) ²A₀ρ(ρD−1)(Q_(S)−A₀D)/{overscore (K)}²

[0095] c3=c2

[0096] c4=C_(K) ²ρ²(Q_(S)−A₀D)²/{overscore (K)}²

[0097] The closed form solution for the integrals the in the aboveequation are $\begin{matrix}{{Term1} = {\frac{1}{4\left( {\rho - {1/D}} \right)^{2}}\left\{ {{\left( {{\rho \quad D} - 1} \right)\left( {{2z^{2}} - {2{zD}} - {D^{2}^{{- 2}{z/D}}} + D^{2}} \right)} +} \right.}} \\{\quad {{\frac{4\left\lbrack {{z\left( {\rho - {1/D}} \right)} + 1} \right\rbrack}{\left( {\rho + {1/D}} \right)^{2}}\left\lbrack {{- {z\left( {\rho + {1/D}} \right)}} - ^{- {z{({\rho + {1/D}})}}} + 1} \right\rbrack} +}} \\{\left. \quad \left\lbrack {{2{zD}} + {D^{2}^{{- 2}{z/D}}} - D^{2}} \right\rbrack \right\} +} \\{\quad {\frac{1}{4\left( {\rho + {1/D}} \right)^{2}}\left\{ {{\left( {{\rho \quad D} + 1} \right)\left( {{2z^{2}} - {2{zD}} - {D^{2}^{{- 2}{z/D}}} + D^{2}} \right)} +} \right.}} \\{\quad {{\frac{4}{\left( {\rho - {1/D}} \right)^{2}}\left\lbrack {{{- {z\left( {\rho - {1/D}} \right)}}^{- {z{({\rho + {1/D}})}}}} + ^{{- 2}{z/D}} - ^{- {z{({\rho + {1/D}})}}}} \right\rbrack} -}} \\\left. \quad \left\lbrack {{2{zD}} + {D^{2}^{{- 2}{z/D}}} - D^{2}} \right\rbrack \right\} \\{{Term2} = {\frac{1}{\rho^{2}}\left\{ {{2{\rho \left( {{z^{2}D} - {2{zD}^{2}} - {2D^{3}^{{- z}/D}} + {2D^{3}}} \right)}} -} \right.}} \\{\quad {{\frac{\left( {1 + {\rho \quad z}} \right)}{\left( {\rho + {1/D}} \right)^{2}}\left\lbrack {{z\left( {\rho + {1/D}} \right)} + ^{- {z{({\rho + {1/D}})}}} - 1} \right\rbrack} +}} \\\left. \quad {\frac{^{{- \rho}\quad z}}{\left( {\rho - {1/D}} \right)^{2}}\left\lbrack {{- {z\left( {\rho - {1/D}} \right)}} + ^{z{({\rho - {1/D}})}} - 1} \right\rbrack} \right\} \\{{Term3} = {\frac{1}{\left( {\rho - {1/D}} \right)^{2}}\left\{ {{\left( {\rho - {1/D}} \right)\left( {{z^{2}D} - {2{zD}^{2}} - {2D^{3}^{{- z}/D}} + {2D^{3}}} \right)} -} \right.}} \\{\quad {{\frac{{z\left( {\rho - {1/D}} \right)} + 1}{\rho^{2}}\left\lbrack {{\rho \quad z} + ^{{- \rho}\quad z} - 1} \right\rbrack} +}} \\{\left. \quad \left\lbrack {{zD} + {D^{2}\quad ^{{- z}/D}} - D^{2}} \right\rbrack \right\} +} \\{\quad {\frac{1}{\left( {\rho + {1/D}} \right)^{2}}\left\{ {{\left( {\rho + {1/D}} \right)\left( {{z^{2}D} - {2{zD}^{2}} - {2D^{3}^{{- z}/D}} + {2D^{3}}} \right)} +} \right.}} \\{\quad {\frac{^{- {z{({\rho + {1/D}})}}}}{\rho^{2}}\left\lbrack {{{- \rho}\quad z} + ^{\rho \quad z} - 1 -} \right.}} \\\left. \quad \left\lbrack {{zD} + {D^{2}\quad ^{{- z}/D}} - D^{2}} \right\rbrack \right\} \\{{Term4} = {\frac{1}{\rho^{2}}\left\{ {{\frac{2}{3}\rho \quad z^{3}} - {\frac{\left( {{\rho \quad z} + 1} \right)}{\rho^{2}}\left( {{\rho \quad z} + ^{{- \rho}\quad z} - 1} \right)} + {\frac{^{{- \rho}\quad z}}{\rho^{2}}\left\lbrack {{{- \rho}\quad z} + ^{\rho \quad z} - 1} \right\rbrack}} \right\}}}\end{matrix}$

[0098] Condition 6: When an exponential heat source function isconsidered and the B.C used are the surface temperature and heat flow atthe base of the model

[0099] The governing heat conduction equation with an exponential heatsource term $\begin{matrix}{{\frac{}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {{- A_{0}}^{{- z}/D}}} & (23)\end{matrix}$

[0100] with constant surface temperature and constant basal heat flow asboundary conditions

[0101] (Type (ii))

[0102] Solution

[0103] Mean Temperature $\begin{matrix}{\overset{\_}{T} = {{E\left( {T(z)} \right)} = {T_{0} + {\frac{Q_{B}}{\overset{\_}{K}}z} + {\frac{A_{0}D^{2}}{\overset{\_}{K}}\left( {1 - {\frac{z}{D}^{{- L}/D}} - ^{{- z}/D}} \right)}}}} & (24)\end{matrix}$

[0104] Variance in the Temperature

σ_(T) ² =c1×Term1+c2×Term2+c3×Term3+c4×Term4  (25)

[0105] where the constants are

[0106] c1=C_(K) ²A₀ ²(1−ρD)²/{overscore (K)}²

[0107] c2=C_(K) ²A₀ρ(ρD−1)(Q_(B)−A₀De^(−L/D))/{overscore (K)}²

[0108] c3=c2

[0109] c4=C_(K) ²ρ²(Q_(B)−A₀De^(−L/D))²/{overscore (K)}²

[0110] The terms Term1, Term2, Term3, Term4 are same as given incondition 5.

[0111] The present invention has its novelty over previous work in thefollowing counts

[0112] 1. The method uses randomness in the thermal conductivitystructure to quantify the errors in the subsurface temperature depthdistribution

[0113] 2. The method used has led to exact closed form solution to themean and its variance on the subsurface temperature field for differentprescribed boundary conditions.

[0114] 3. The present solutions will be used extensively for quantifyingthe subsurface temperatures for any given region. The exact formulae formean and variance in the subsurface temperature depth distribution havenot been given so far and have a wide application in geothermal studies.

[0115] The following examples are given by way of illustrations andtherefore should not be constructed to limit the scope of the presentinvention.

EXAMPLE 1

[0116] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 1 Boundary conditions: SurfaceTemperature (T₀)   0 (° C.) Surface heat flow (Q_(s))  80 (mW/m²) ModelDepth (L). 3.5 (km) Random thermal conductivity: Mean thermalconductivity {overscore (K)} 3.0 (mW/m²) Coefficient of variabilityC_(k) 0.4 Correlation length scale 1/ρ   1 km

[0117] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (9) and (10) andthe results have been plotted in FIG. 1. From the figures we see thatthe error bounds on the temperature increases with depth, increases withan increase in the coefficient of variability in thermal conductivityand in the correlation length scale

EXAMPLE 2

[0118] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 2 Boundary conditions: SurfaceTemperature (T₀)   0 (° C.) Basal heat flow (Q_(B))  30 (mW/m²) ModelDepth (L). 5.5 (km) Random thermal conductivity: Mean thermalconductivity {overscore (K)} 2.0 (mW/m²) Coefficient of variabilityC_(k) 0.3 Correlation length scale 1/ρ   1 km

[0119] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (12) and (13)and the results have been plotted in FIG. 2. From the figures we seethat the error bounds on the temperature increases with depth, increaseswith an increase in the coefficient of variability in thermalconductivity and in the correlation length scale

EXAMPLE 3

[0120] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 3 Boundary conditions: SurfaceTemperature (T₀)  30 (° C.) Surface heat flow (Q_(s))  40 (mW/m²) HeatSource Radiogenic heat production (A) 2.5 (μW/m³) Model Depth (L).  10(km) Random thermal conductivity: Mean thermal conductivity {overscore(K)} 3.0 (mW/m²) Coefficient of variability C_(k) 0.2 Correlation lengthscale 1/ρ   3 km

[0121] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (15) and (16)and the results have been plotted in FIG. 3. From the figures we seethat the error bounds on the temperature increases with depth, increaseswith an increase in the coefficient of variability in thermalconductivity and in the correlation length scale.

EXAMPLE 4

[0122] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 4 Boundary conditions: SurfaceTemperature (T₀)  30 (° C.) Surface heat flow (Q_(B))  20 (mW/m²) HeatSource Radiogenic heat production (A) 2.5 (μW/m³) Model Depth (L).  10(km) Random thermal conductivity: Mean thermal conductivity {overscore(K)}  3. (mW/m²) Coefficient of variability C_(k) 0.5 Correlation lengthscale □   4 km

[0123] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (18) and (19)and the results have been plotted in FIG. 4. From the figures we seethat the error bounds on the temperature increases with depth, increaseswith an increase in the coefficient of variability in thermalconductivity and in the correlation length scale.

EXAMPLE 5

[0124] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 5 Boundary conditions: SurfaceTemperature (T₀)  30 (° C.) Surface heat flow (Q_(s))  43 (mW/m²) HeatSource Radiogenic heat production (A) 2.6 (μW/m³) Characteristic Depth(D)  12 (km) Model Depth (L).  35 (km) Random thermal conductivity: Meanthermal conductivity {overscore (K)} 3.0 (mW/m²) Coefficient ofvariability C_(k) 0.3 Correlation length scale 1/ρ  10 km

[0125] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (21) and (22)and the results have been plotted in FIG. 5. From the figures we seethat the error bounds on the temperature increases with depth, increaseswith an increase in the coefficient of variability in thermalconductivity and in the correlation length scale.

EXAMPLE 6

[0126] Numerical values of the controlling input thermal parameters fora Realistic Earth model for condition 6 Boundary conditions: SurfaceTemperature (T₀)   0 (° C.) Surface heat flow (Q_(B))  20 (mW/m²) HeatSource Radiogenic heat production (A) 2.2 (μW/m³) Characteristic Depth(D)  10 (km) Model Depth (L).  35 (km) Random thermal conductivity: Meanthermal conductivity {overscore (K)} 2.6 (mW/m²) Coefficient ofvariability C_(k) 0.2 Correlation length scale 1/ρ  11 km

[0127] Using these controlling thermal parameters the mean temperatureand its error bounds have been computed using equations (24) and (25)and the results have been plotted in FIG. 6. From the figures we seethat the error bounds on the temperature increases with depth, increaseswith an increase in the coefficient of variability in thermalconductivity and in the correlation length scale.

[0128] The Main Advantages of the Invention are:

[0129] 1. The advantage of this invention is that exact formulaes havebeen given to quantify the error bounds on the subsurface temperaturesdue to errors in the thermal conductivity for a conductive earth model.

[0130] 2. The errors in the temperatures will help in a betterevaluation of the crustal thermal structure.

[0131] 3. This study can be used in quantifying the conductive thermalstructure along with its error bounds for any given region and therebyhelp in understanding the geodynamics of the region.

[0132] 4. The exact formulae for the mean temperature and its standarddeviation can be used in a better evaluation of the thermal state of oilbearing regions. These exact solutions can be used in tectonic studiesand in studies related to crystallization of minerals.

REFERENCES

[0133] Gallagher, K., Ramsdale, M., Lonergan. L., and Marrow, D., 1997,The role thermal conductivities measurements in modeling the thermalhistories in sedimentary basins, Mar. Petrol. Geol., 14, 201-214.

[0134] Jokinen. J. and Kukkonen. I. T., 1999a, Random modeling oflithospheric thermal regime: Forward simulation applied in uncertaintyanalysis, Tectonophysics, 306, 277-292.

[0135] Jokinen. J. and Kukkonen. I. T., 1999b, Inverse simulation oflithospheric thermal regime using the Monte Carlo method, 306, 293-310.

[0136] Nielson, S. B., 1987, Steady state heat flow in a random mediumand linear heat flow heat production relationship, Geophys. Res. Lett.14, 318-321.

[0137] Royer J. J. and Danis, M., 1988, Steady state geothermal model ofthe crust and problems of boundary conditions: Application to a riftsystem, the southern Rhinegraben, Tectonophysics, 156, 239-255.

[0138] Serrano, S. E., 1995, Forecasting scale dependent dispersion fromspills in heterogeneous aquifers, J. Hydrology, 169, 151-169.

[0139] Srivastava, K., and Singh, R. N., 1998, A model for temperaturevariation in sedimentary basins due to random radiogenic heat sources,Geophys. J. Int., 135, 727-730.

[0140] Srivastava, K. and Singh, R. N., 1999, A stochastic model toquantify the steady state crustal geotherms subject to uncertainty inthermal conductivity, Geophy. J. Int, 138, 895-899.

[0141] Vasseur, G., and Singh, R. N., 1986, Effect of random horizontalvariation in radiogenic heat source distribution on its relationshipwith heat flow, J. Geophys. Res. 91, 10397-10404.

[0142] Vasseur G., Lucazeau. F. and Bayer, R., 1985, The problem of heatflow density determination from inaccurate data, Tectonophysics, 121,23-34.

We claim:
 1. A method for obtaining closed form expressions forsubsurface temperature depth distribution along with its error bounds,the method comprising using a stochastic heat conduction equationincorporating random thermal conductivity to obtain a mean and variancein temperature fields for a set of boundary conditions: the equationconsisting of $\begin{matrix}{{\frac{}{z}\left\{ {\left( {\overset{\_}{K} + {K^{\prime}(z)}} \right)\frac{T}{z}} \right\}} = {- {A(z)}}} & (1)\end{matrix}$

where T is the temperature (° C.) A(z) is the radiogenic heat source (μW/m³), K(z)={overscore (K)}+K′(z) is the thermal conductivity (W/m ° C.)which is expressed as a sum of a deterministic component and a randomcomponent K′(z) is the random component with mean zero and a Gaussiancolored noise correlation structure represented by E(K′(z))=0  (2)E(K′(z ₁)K′(z ₂))=σ_(K) ² e ^(−ρ|z) ^(₁) ^(−z) ^(₂) ^(|)  (3) whereσ_(K) ² is the variance is thermal conductivity (W/m ° C.)² ρis thecorrelation decay parameter m⁻¹ (or 1/ρ is the correlation length scale)and z₁ and Z₂ are the depths (m)
 2. A method as claimed in claim 1wherein the boundary condition consists of condition of heat sources andis selected from the group consisting of Zero (A(z)=0), Constant(A(z)=A) and exponentially decreasing with depth (A(z)=A₀e^(−z/D)).
 3. Amethod as claimed in claim 1 wherein the boundary condition comprisesconstant surface temperature and constant surface heat flow.
 4. A methodas claimed in claim 1 wherein the boundary condition comprises constantsurface temperature and constant basal heat flow.
 5. A method as claimedin claim 1 wherein a parameter used is that of radiogenic heatgeneration.
 6. A method as claimed in claim 1 wherein the method iscarried out electronically using a computing means and whereinappropriate numerical values are given for controlling thermalparameters directly in the boxes that appear on the screen, therebyinstantaneously computing and plotting the mean and error bounds on thetemperature depth distribution.
 7. A method as claimed in claim 1wherein the subsurface is selected from an oil field, a natural gasfield, tectonically active area and a mineral resource area.